On the ill-posed Cauchy problem for the polyharmonic heat equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912201567109120 |
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| author | Kurilenko, Ilya Shlapunov, Alexander |
| author_facet | Kurilenko, Ilya Shlapunov, Alexander |
| contents | We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- Δ)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$, where $n\geq 1$, $m\geq 1$ and $Δ$ is the Laplace operator, via its values and the values of its normal derivatives up to order $(2m-1)$ on a given part of the lateral surface of the cylinder. We obtain a Uniqueness Theorem for the problem and a criterion of its solvability in terms of the real-analytic continuation of parabolic potentials, associated with the Cauchy data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07797 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the ill-posed Cauchy problem for the polyharmonic heat equation Kurilenko, Ilya Shlapunov, Alexander Analysis of PDEs 35K25, 35R25 We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- Δ)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$, where $n\geq 1$, $m\geq 1$ and $Δ$ is the Laplace operator, via its values and the values of its normal derivatives up to order $(2m-1)$ on a given part of the lateral surface of the cylinder. We obtain a Uniqueness Theorem for the problem and a criterion of its solvability in terms of the real-analytic continuation of parabolic potentials, associated with the Cauchy data. |
| title | On the ill-posed Cauchy problem for the polyharmonic heat equation |
| topic | Analysis of PDEs 35K25, 35R25 |
| url | https://arxiv.org/abs/2212.07797 |